document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=531437>Question 531437</A>: <I><SPAN STYLE=\"word-break: break-all;\">   ______8______          <BR>\n" );
document.write( "  |_|x             <BR>\n" );
document.write( "  |x<BR>\n" );
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document.write( "13|<BR>\n" );
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document.write( "This is a paper that is cut from EACH side in small squares when you fold it is a rectangular prism with no top, 13 is the lenght and 8 is the width. it did'nt let me draw the other half but it is a FULL rectangle and all four sides of the corners a cube is cut out and that is X.  What is the value of X and Maximum volume of it and could you show work too my Teacher said use a TI calculator if you can and i dont have one and she said that it was going to be a parabola it would be great if you could help me thanks!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #350482 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=350482\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Once the box is folded, the length will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 13\ -\ 2x\">, the width will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 8\ -\ 2x\">, and the height will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  Since the volume of a rectangular prism is given by length times width times height, we create the volume function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ (13\ -\ 2x)(8\ -\ 2x)(x)\ =\ 4x^3\ -\ 42x^2\ +\ 104x\">\r<BR>\n" );
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document.write( "The graph of which is NOT a parabola, although the portion of the graph of interest, that is where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> is within an interval that allows for an actual physical box to be created, sorta-kinda <i>looks</i> like a parabola.\r<BR>\n" );
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document.write( "In order to actually create a box, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> must be in the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(0,\,4)\">.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> were negative you would be dealing with the absurdity of cutting a negative amount from the corners.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0\">, then you aren't cutting anything away from the corners and all you have is a flat 13 by 8 piece of paper with zero volume.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 4\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 8\ -\ 2x\ =\ 8\ -\ 2(4)\ =\ 0\"> and again, you have zero volume.\r<BR>\n" );
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document.write( "Somewhere in between is a maximum volume, and we know this because of the Mean Value Theorem.\r<BR>\n" );
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document.write( "If a function is continuous and twice differentiable over an interval and if the first derivitive of the function is equal to zero at some point in the interval, then there is a local extremum at that point.  If the second derivitive is negative at that point, then the extremum is a maximum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ 12x^2\ -\ 84x\ +\ 104\">\r<BR>\n" );
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document.write( "Set the derivitive equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 12x^2\ -\ 84x\ +\ 104\ =\ 0\">\r<BR>\n" );
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document.write( "Solve this quadratic and select the value that is within the previously discussed interval.  This is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> value that gives the extreme volume.  Evaluate the original volume function for this value to get the actual extreme volume.\r<BR>\n" );
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document.write( "Take the second derivitive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ 24x\ -\ 84\">\r<BR>\n" );
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document.write( "and evaluate it at the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> that gives the extreme volume.  The second derivitive must be negative if the extreme point is a maximum.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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